Reduced phase space quantization

Abstract

We examine two singular Lagrangian systems with constraints which apparently reduce the phase space to a 2-dimensional sphere and a 2-dimensional hyperboloid. Rigorous constraint analysis by Dirac's method, however, gives 2-dimensional open disc and an infinite plane with a hole in the centre respectively as the reduced phase spaces. Upon canonical quantisation the classical constraints show up as restrictions on the Hilbert space.